Adding Poisson Distributions | 独立泊松分布的可加性

📚 Adding Poisson Distributions | 独立泊松分布的可加性

In Edexcel A-Level Statistics, one of the most useful properties of the Poisson distribution is that the sum of two independent Poisson variables is also Poisson. If X ~ Po(λ) and Y ~ Po(μ) are independent, then X + Y ~ Po(λ + μ). This rule lets us combine independent counts from different sources and work with a single Poisson model.

在 Edexcel A-Level 统计中，泊松分布最有用的性质之一是：两个独立泊松变量之和仍服从泊松分布。若 X ~ Po(λ) 且 Y ~ Po(μ) 独立，则 X + Y ~ Po(λ + μ)。这一法则使我们可以把来自不同来源的独立计数合并，并使用单一的泊松模型进行分析。

1. Quick Recap: The Poisson Distribution | 快速回顾：泊松分布

A discrete random variable X follows a Poisson distribution with parameter λ if its probability mass function is given by:

如果离散随机变量 X 服从参数为 λ 的泊松分布，其概率质量函数为：

P(X = x) = e^(−λ) λˣ / x! for x = 0, 1, 2, …

The parameter λ represents the mean number of occurrences in a fixed interval. The mean and variance of a Poisson variable are both equal to λ.

参数 λ 表示固定区间内事件发生的平均次数。泊松变量的均值和方差都等于 λ。

E(X) = λ and Var(X) = λ

2. The Addition Rule for Two Poisson Variables | 两个泊松变量的加法法则

Let X ~ Po(λ) and Y ~ Po(μ), where X and Y are independent. Then the sum S = X + Y has a Poisson distribution with parameter λ + μ:

设 X ~ Po(λ)，Y ~ Po(μ)，且 X 与 Y 相互独立。则和 S = X + Y 服从参数为 λ + μ 的泊松分布：

S = X + Y ~ Po(λ + μ)

This is often called the addition property or reproductive property of the Poisson distribution. It holds only for independent variables, not for any two random variables with Poisson marginal distributions.

这通常称为泊松分布的加法性质或再生性质。它只对独立变量成立，并不是任意两个边缘分布为泊松的变量之和都服从泊松分布。

3. Key Conditions: Independence and Same Interval | 关键条件：独立性与相同区间

For the addition rule to be valid, the following conditions must be satisfied:

要使加法法则有效，必须满足以下条件：

• X and Y must be independent.
• Both variables must count events over the same fixed interval, area, volume, or region, unless parameters are adjusted for exposure.
• No constant scaling is applied to the variables.

X 与 Y 必须相互独立；两个变量必须在相同的固定区间、面积、体积或区域内计数，除非已对暴露量调整参数；变量没有经过常数倍缩放。

A common error is to write 2X ~ Po(2λ). This is false because multiplying a Poisson variable by 2 is not the same as adding two independent Poisson variables with the same parameter. The variable 2X takes only even values, so it cannot be Poisson.

常见错误是把 2X 写成 Po(2λ)。这是错误的，因为把泊松变量乘以 2 并不等价于两个同参数独立泊松变量相加。变量 2X 只能取偶数值，因此不可能是泊松分布。

4. Why the Parameters Add: Intuition and Proof Sketch | 参数为何相加：直观理解与证明思路

Intuition: if X counts events from one independent source at rate λ and Y counts events from another independent source at rate μ, then merging the two sources produces a single Poisson process with rate λ + μ.

直观理解：如果 X 计数来自一个独立源、速率为 λ 的事件，Y 计数来自另一个独立源、速率为 μ 的事件，那么合并两个源会得到一个速率为 λ + μ 的泊松过程。

A formal check uses the probability generating function. For X ~ Po(λ), the probability generating function is G_X(t) = E(t^X) = e^(λ(t−1)). Since X and Y are independent:

正式验证可使用概率生成函数。对于 X ~ Po(λ)，概率生成函数为 G_X(t) = E(t^X) = e^(λ(t−1))。由于 X 与 Y 独立：

G_{X+Y}(t) = G_X(t) G_Y(t) = e^(λ(t−1)) e^(μ(t−1)) = e^((λ+μ)(t−1))

The resulting generating function is exactly the probability generating function of a Poisson distribution with parameter λ + μ, so the sum is Poisson.

得到的生成函数正是参数为 λ + μ 的泊松分布的概率生成函数，因此和服从泊松分布。

5. Extending to Three or More Poisson Variables | 推广到三个或更多泊松变量

The addition property extends naturally. If X₁, X₂, …, Xₙ are independent Poisson variables with parameters λ₁, λ₂, …, λₙ, then:

加法性质可以自然推广。若

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